Unveiling the Velocity-Space Signature of Ion Cyclotron Damping Using Liouville Mapping

TL;DR

The paper addresses how ion cyclotron damping transfers energy from waves to ions in collisionless plasmas by developing a computationally efficient framework that couples Liouville mapping with field-particle correlations (FPC). By constructing prescribed wave fields from PLUME eigenfunctions and tracing single-particle trajectories backward in time, the authors obtain the velocity-space distribution and compute FPCs to reveal the secular energization patterns. They validate the approach by reproducing the Landau-damping signatures for kinetic Alfvén waves and then uncover the ion cyclotron damping signatures: a quadrupolar pattern in the perpendicular velocity plane and a localized energization near the $n=1$ resonant velocity in the gyrotropic $(v_\parallel,v_\perp)$ space, with only weak dependence on $\beta_i$. The method’s efficiency enables systematic β-scale exploration and provides a practical foundation for identifying ion cyclotron damping in kinetic simulations and spacecraft data, with data and code openly available. These findings offer a robust diagnostic tool for parsing wave–particle interactions in turbulent plasmas and contribute to the broader understanding of collisionless plasma heating.

Abstract

Ion cyclotron damping is a key mechanism for the dissipation of electromagnetic wave energy in weakly collisional plasmas. This study presents a combined approach using Liouville mapping and the field-particle correlation technique to investigate qualitatively and quantitatively the velocity-space signature of ion cyclotron damping. Liouville mapping offers a computationally efficient way to predict perturbations to the particle velocity distribution function using single-particle trajectories in prescribed electromagnetic fields. One may apply the field-particle correlation technique to these perturbed velocity distributions to reveal the unique velocity-space signatures of the secular energy transfer rate associated with specific wave-particle interactions. We validate this method by reproducing known Landau damping signatures for kinetic Alfvén waves, and then we apply this method to ion cyclotron waves where ion cyclotron damping dominates. The resulting velocity-space signature reveals distinct energization features of ion cyclotron damping : (i) a quadrupolar pattern in the perpendicular $(v_x, v_y)$ plane; and (ii) a localized energization near the $n = 1$ resonant velocity in gyrotropic $(v_\parallel, v_\perp)$ velocity-space. The quantitative patterns remain unchanged as the ion plasma beta $β_i$ is varied, ultimately showing minimal $v_\perp$ dependence on $β_i$ of the velocity-space signature at the $n = 1$ resonant velocity. This work provides a systematic study of how the ion cyclotron damping signature varies with $β_i$, offering a practical foundation to identify ion cyclotron damping using kinetic simulation data or spacecraft data.

Figures (13)

• Figure 1: Illustration of the procedure for Liouville mapping of the velocity distribution function by integrating the single-particle motion (SPM) equations backwards in time from final state $(\mathbf{r}_0, \mathbf{v}, t_f)$ back to initial state $(\mathbf{r}_\text{init}, \mathbf{v}_\text{init}, t_\text{init})$. This procedure must be completed for each of the $M$ points in the 3D velocity-space grid, and for each of the $N$ subintervals that span the correlation interval $\tau$.
• Figure 2: Flowchart describing the full procedure to construct the electromagnetic fields, perform Liouville mapping of the distribution function, and computation of the FPC, as described in Sec. \ref{['sec:lm']}. Here, "instantaneous FPCs" refer to the integrands in the definitions of FPCs, i.e. Eq. (\ref{['eq:cex']})-(\ref{['eq:ceperp']}), and "rk45" denotes the Runge-Kutta-Fehlberg method.
• Figure 3: The normalized (a) frequency $\omega / (k_\parallel v_A)$ and (b) damping rates $|\gamma|/\omega$ for a KAW in plasma with parameters $\beta_i = 1$, $T_i/T_e = 1$, $m_i/m_e = 1836$, $v_\text{ti} / c = 10^{-4}$. The KAW wave mode with $k_\parallel \rho_i = 0.05$ is plotted over the range $10^{-1} \leq k_\perp \rho_i \leq 10^1$. The damping rates plotted include the total damping rate $|\gamma|/\omega$ (black), ion damping rate $|\gamma_i|/\omega$ (red solid), electron damping rate $|\gamma_e|/\omega$ (blue), and damping rate due to only ion Landau damping $|\gamma_{i,LD}|/\omega$ (red dashed).
• Figure 4: The gyrotropic parallel FPC $C_{E_\parallel}(v_\parallel,v_\perp)$ (top panels) and reduced parallel FPC $C_{E_\parallel}(v_\parallel)$ (bottom panels) for (a,b) the forward-propagating KAW case, (c,d) the backward-propagating KAW case, and (e,f) the counterpropagating KAW case, showing bipolar velocity-space signatures arise in the directions corresponding to the wave propagation direction, as expected theoretically.
• Figure 5: Linear dispersion relation solution for an ICW from PLUME for plasma parameters $\beta_i = 1$, $T_i/T_e = 1$, $v_{ti}/c = 10^{-4}$, and $m_i / m_e = 1836$. We set the perpendicular component of the wavenumber to $k_\perp \rho_i = 0.01$ and present solutions for (a) normalized wave frequency $\omega/\Omega_i$ (black), (b) electric field polarization $\mathcal{P}_E$, and (c) normalized damping rates $|\gamma|/\omega$ as a function of $k_\parallel \rho_i$. The contributions of ion Landau damping (iLD), ion transit-time damping (iTTD), and ion cyclotron damping (iCD) are separately plotted in (c), showing the rapid and strong onset of ion cyclotron damping as $k_\parallel \rho_i \rightarrow 1$. An empirical analytical expression for the cold plasma ICW wave frequency $\omega/\Omega_i$ (red dashed), given by Eq. (\ref{['eq:cp_icw']}), is plotted in (a) for comparison.